Intuition The one core ideaElectric charge fills the space around it with invisible "push arrows" (a field), and if you draw an imaginary closed bag around some charge, the total number of arrows poking out of that bag depends only on how much charge sits inside it. When the bag's shape matches the charge's symmetry, counting those arrows becomes simple multiplication — and that is the entire engine behind Gauss's Law .
This page builds every symbol used in the parent note Applications — sphere, cylinder, infinite plane starting from nothing. If the parent wrote a symbol without explaining it, you will find it explained here first.
Definition Electric chargeA property some particles carry that makes them push or pull on each other. Measured in coulombs , symbol C Q q
Picture: a glowing dot. Positive charge = a source that shoots arrows outward ; negative charge = a sink that swallows arrows inward .
Why the topic needs it: every formula in the parent starts with "how much charge?" — a sphere holds Q
The arrows in the picture are the electric field , which we meet next.
Definition Electric fieldE E arrow telling you which way (and how hard) a positive test charge would be pushed if placed there. The little arrow on top,
Size (magnitude) is written plain, E N/C Direction is where the arrow points.
arrow and not just a number?
A push is useless unless you know which way it pushes. A number alone (5 N/C vector , and the hat
Why the topic needs it: E
The parent uses three different distance letters. They are not interchangeable — mixing them is the #1 slip.
Definition The distance symbols
R fixed size of the object itself: the radius of the sphere, the radius of the cylinder. It never changes during a problem.r centre of a sphere out to the point where you want the field. It varies.s axis of a cylinder/line straight out sideways to your point. It varies.
Intuition Why two different small letters?A sphere is symmetric about a point , so distance is measured from that point → r line , so distance is measured straight out from that line → s r
Pointing straight away from the centre (sphere) or away from the axis (cylinder). Look at the picture: every arrow lies along the ruler line, never sideways.
Why the topic needs it: the whole trick is that symmetry forces E only on r s
Real charge is rarely a single dot; it is smeared over a length, an area, or a volume. "Density" means "how much charge per unit of that thing."
Definition The three densities
λ linear charge density: coulombs per metre of length, C/m σ surface charge density: coulombs per square metre, C/m 2 ρ volume charge density: coulombs per cubic metre, C/m 3
Intuition Why we need densities at allTo find the charge inside your imaginary bag, you multiply the density by how much of the object the bag contains:
length inside → Q = λ × L
area inside → Q = σ × A
volume inside → Q = ρ × 3 4 π r 3
These are exactly the Q e n c
Common mistake Confusing which density goes with which shapeFix: count the dimensions. A line is 1-D → λ σ ρ
Definition The area elementd A size (its area, d A direction : the arrow d A out of the patch, perpendicular to it, pointing outward from the enclosed region.
The d
Picture: a tiny tile on a curved surface with a pin poking straight out of its face.
Why the topic needs it: to measure how many field arrows pierce a patch, we compare the field arrow E d A
Definition Dot product of two arrowsE ⋅ d A = E d A cos θ θ
If they point the same way (θ = 0 cos 0 = 1 E d A
If they are perpendicular (θ = 9 0 ∘ cos 9 0 ∘ = 0 0 along the surface, pierces nothing.
Intuition Why the dot product and not plain multiplication?We want "how much field goes through the patch," not how much slides along it. Only the part of E d A zero — cos 9 0 ∘ = 0
See Electric Flux for this concept as its own topic.
∮ S ∮ ∫ closed surface" — a surface with no holes, like a balloon skin that fully wraps a region.
∮ S E ⋅ d A = (add E ⋅ d A over every patch of the closed bag S )
Picture: counting every field arrow poking out through the whole balloon, with arrows poking in counted as negative.
Gauss's law relates arrows-out to charge-inside . "Inside" only makes sense if the surface fully seals off a region — an open sheet has no inside. That is why we always use a fully sealed Gaussian surface .
Q e n c inside your chosen Gaussian surface — and only that. Charge outside the bag contributes nothing to the net arrows out.
ε 0 permittivity of free space , a fixed constant of the universe:
ε 0 = 8.85 × 1 0 − 12 C 2 / ( N ⋅ m 2 ) k = 4 π ε 0 1 = 9 × 1 0 9 N ⋅ m 2 / C 2 Coulomb's Law ).
ε 0 1/ ε 0 ε 0 ε 0
Putting the pieces together gives the master tool the parent opens with:
∮ S E ⋅ d A = ε 0 Q e n c
Definition Symmetry (here)A shape has symmetry when it looks identical after you move/turn/flip it. A sphere looks the same after any rotation about its centre; an infinite line looks the same as you slide along it or spin around it; an infinite sheet looks the same everywhere you stand.
Intuition Why symmetry is the master keyGauss's law is always true but usually useless, because E E must have the same magnitude on the matching surface and point straight along d A ∮ E × A Symmetry in Physics .
Source symmetry
Matching Gaussian surface
Result power law
Spherical (point-like)
Concentric sphere
E ∝ 1/ r 2
Axial (line/cylinder)
Coaxial cylinder
E ∝ 1/ s
Planar (sheet)
Pillbox
E =
Definition Conductor in electrostaticsA material whose charges can move freely. Once settled (electrostatic equilibrium): the field inside the metal is zero, and all excess charge sits on the outer surface . Full topic: Conductors in Electrostatics .
Why the topic needs it: this is why "inside a conducting shell, E = 0 Q e n c = 0 E = σ / ε 0 σ /2 ε 0 Parallel Plate Capacitor and Electric Potential chapters ahead.
Densities rho lambda sigma
Closed integral total flux
E constant so E times area
Gauss law E times area equals Q enc over eps0
Sphere Cylinder Plane results
Test yourself — reveal only after answering.
What does the hat in E That
E is a vector — it carries a direction, not just a size.
Difference between r s r s
What is R The fixed radius of the object itself (sphere or cylinder); it does not vary during the problem.
Which density goes with a wire, a sheet, a solid ball? Wire → λ σ ρ
Which way does d A Straight out of the surface patch, perpendicular to it, outward from the enclosed region.
When is E ⋅ d A = 0 When
E is perpendicular to
d A (
θ = 9 0 ∘ ,
cos 9 0 ∘ = 0 ) — field skims along the surface.
Why the circle on ∮ It means the integral is over a closed surface, so "inside" is well-defined.
What does Q e n c Only the charge inside your chosen Gaussian surface — outside charge is irrelevant.
Value and meaning of ε 0 8.85 × 1 0 − 12 C 2 / ( N⋅m 2 )
Why does symmetry make Gauss solvable? It forces
E to be constant on the matching surface and parallel to
d A , so
∮ collapses to
E × A .
Field inside a conductor at equilibrium? Zero; all excess charge sits on the outer surface.
